Analyzing energy loss in a capacitor circuit

We often run into the situation where there are two capacitors, one of them charged up to some voltage level and the other at some lower voltage level which, for purposes of this discussion, we will say means discharged to zero volts. Then, we close some switch connected as shown in Figure 1.

Figure 1 Diagram of a circuit switching two capacitors together where the voltage at t = 0 is Va for C1 and 0 for C2. Then the switch is closed and, as time moves to infinity, both C1 and C2 will be charged to voltage Vb. Source: John Dunn

If we wait until the end of time, the voltages of the two capacitors will eventually become equal to each other. If on the other hand, we only wait for some reasonable time lying within our personal life spans, the two capacitors’ voltages will get pretty close to each other. How long that process will seem to take will be a function of the circuit’s time constant which will itself be some value that is proportional to the resistance value of “R”.

There will be some loss of energy during the described process which we can examine in two ways, first by the conservation of charge (Figure 2) and then by calculus (Figure 3).

Figure 2 First analysis with the conservation of charge where the energy loss is worked out algebraically.

Figure 3 Second analysis yields the same result as the first and is done with calculus where total energy loss is equivalent to the resistor energy and is ultimately derived from the resistor’s power.

The semi-obvious conundrum which someone might notice is that there has been a loss of energy in a circuit in which if the value of R goes to zero and yet there is still a non-zero energy loss. Therefore, where does that lost energy go?

Someone once suggested to me that energy loss for R = 0 comes about via the magical radiation of a pulse of electromagnetic energy. Say “no” to that idea. The two energy loss analyses yield the same result, which is that given unlimited time, the energy loss is of the amount shown and is invariant with respect to the value of resistance, R.

If we allow R to go to zero, the energy loss remains constant, but it happens more quickly which means the event’s power level rises, heading for infinity, as the event’s time duration shrinks while R is heading for zero. Through all that, the energy loss remains constant (Figure 4).

Figure 4 A statement of the final revealed truth.

John Dunn is an electronics consultant, and a graduate of The Polytechnic Institute of Brooklyn (BSEE) and of New York University (MSEE).

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